Ebert et al Size distributions with pulsed recruitment 



241 



model such as Eq. 6. The changing shapes of the size- 

 frequency distributions for a species with pulsed re- 

 cruitment could be summed and so be made to ap- 

 proximate the shape that would be obtained with 

 continuous recruitment. To obtain a reasonable approxi- 

 mation, it would be necessary to ( 1) take many evenly- 

 spaced samples between recruitment events, and 

 (2) weight the samples with the survival rate, e~ Zt , 

 from the time of recruitment, t. Weighting could be 

 accomplished if accurate estimates of density were 

 known, which, of course, would be the same as know- 

 ing survival. An obvious variant would be the case in 

 which the same area was sampled each time and all 

 individuals were measured. Such a procedure would 

 result in the largest N for the sample immediately 

 following recruitment and the smallest TV for the sample 

 just prior to the next recruitment episode. All samples 

 would be pooled before the size-frequency distribution 

 would be constructed. 



Approximation of a species with pulsed recruitment 

 to a continuous form could be the same as distribu- 

 tions shown in Figs. 2 and 3. For example, if a species 

 had a single pulse of recruitment and was sampled 10 

 evenly-spaced times during a year, and each sample 

 was weighted according to the survival rate, then the 

 summed frequency distribution would be C in Figs. 2 

 and 3. However, if size data were gathered in such a 

 manner that weighting was not automatic, survival 

 rate would have to be obtained by some other tech- 

 nique before size distributions could be summed to 

 approximate continuous recruitment. Techniques for 

 obtaining survival rate, the weighting factor, from size 

 data include those presented by Ebert (1973, 1987), 

 Saila & Lough (1981), Fournier & Breen (1983), and 

 Pauly ( 1987). It must be noted that analysis of a series 

 of size distributions to obtain the weighting factor 

 would provide information on growth as well as sur- 

 vival, and so there would be scant motivation for con- 

 structing a summed distribution. 



It is not possible to infer the causes underlying an 

 observed size distribution from a single sample or even 

 from several samples that are widely spaced in time. 

 For example, bimodal size distributions can arise from 

 intra-cohort (e.g., Shelton et al. 1979, Timmons et al. 

 1980) or inter-cohort (Johnson 1976) competition, and 

 the simple models examined here and in Barry & 

 Tegner (1990) demonstrate that similar size distribu- 

 tions can result from very different mechanisms. In a 

 time-series of size distributions, when the smallest 

 mode shifts through time, the simplest explanation for 

 bimodality is pulsed recruitment (e.g., McPherson 1965, 

 Hickman 1979, Dafni & Tobol 1986/87, Davoult et al. 

 1990). If sampling is adequate and the smallest mode 

 of a bimodal distribution does not shift during the year 

 (e.g., Gladfelter 1978), the most probable explanation 



is continuous recruitment coupled with high mortality 

 rates for the smallest animals and improved survival 

 with increased size, which is a case that fits the expla- 

 nation for bimodality provided by Barry & Tegner 

 (1990). When size distributions are bi- or polymodal 

 and are presented without a time-series (e.g., Tegner 

 & Dayton 1981, Stein & Pearcy 1982, Wilson 1983), 

 reasonable hypotheses can be formulated, but testing 

 requires additional data. 



There are numerous examples of pulsed recruitment 

 for sea urchins in California. Size-frequency distribu- 

 tions gathered for purple sea urchins at Papalote Bay, 

 Baja California, Mexico (31°42") (Pearse et al. 1970) 

 may indicate multiple settlement events each year dur- 

 ing 1962-69 because samples from January, April, and 

 June-November all had a mode < 1.0 cm (Pearse et al. 

 1970, Ebert 1983). However, if growth was very slow 

 at Papalote Bay, as also indicated by the size-frequency 

 distributions, a single settlement episode would ex- 

 plain the data because individuals with a mode at 

 0.5 cm were observed only in summer and fall samples. 



Published size data for sea urchins at Whites Point 

 (33°43'N) and Point Vicente (33°44'N) during 1966 and 

 1967 (Pearse et al. 1970) show recruitment pulses for 

 both species of Strongylocentrotus and for Lytechinus. 

 Recruitment was better in 1966 than in 1967, and 

 small individuals were collected in September 1966 as 

 well as in July and August 1967. Recruitment was not 

 continuous at either Whites Point or at Point Vicente. 



Finally, our results showing pulsed settlement for 

 red and purple sea urchins corroborate the observa- 

 tion of a single spike of settlement at Naples Reef 

 (34°25'N) off Santa Barbara in May 1986 (Rowley 1989) 

 and the report by Harrold et al. (1991) of two recruit- 

 ment events during a year in central California. The 

 pulsed nature of recruitment means that analysis of 

 size-frequency distributions of Strongylocentrotus spp. 

 should not be based on a model that explicitly requires 

 continuous and constant recruitment (Eq. 6). 



We have demonstrated that by using fixed growth 

 and survival parameters, it is possible to generate a 

 wealth of size-distribution shapes merely by changing 

 the number of recruitment episodes/yr. We have inten- 

 tionally focused on this aspect of size-distribution shape 

 because we believe that it forms the stumbling block 

 to the application of the Barry & Tegner model. In 

 effect, their model does not provide a convenient way 

 of gaining insight into demographics because, in order 

 to use it to divine the relative magnitude of param- 

 eters, it would be necessary to demonstrate the pat- 

 tern of recruitment for the population being studied. 

 Since the preponderance of field evidence indicates that 

 recruitment generally is pulsed, one cannot "...draw 

 inferences concerning the demographic dynamics of a 

 population. ..simply by observing the shape of its size- 



